I've been assigned to teach our undergraduate course in geometry next semester. This course originally was intended for future high-school teachers and focused on axiomatic, Euclid-style geometry (planar, spherical, and hyperbolic). Rice University has changed a lot since this course began being taught (many, many years ago); we now have very few students who want to be high school teachers, and in general the level of our students is such that most of our math majors perceive the course to be beneath them.

My assignment is to redesign the course. I have almost complete freedom except that I cannot require any prerequisites beyond multivariable calculus and ODE's.

Question : What textbook should I use?

Here are my thoughts about what I am looking for.

As I said, I cannot require any prerequisites beyond multivariable calculus and ODE's. However, our undergraduate students are very strong (based on test scores and high school grades, they are pretty similar to the students at eg Cornell or Northwestern). So I want a book that has plenty of meat in it.

It should contain a mixture of proofs and computation, but plenty of proofs.

There are no topics that I am required to cover, though of course it has to be geometric (in particular, this course is not a prerequisite for anything else).

I find axiomatic treatments of geometry boring.

I don't want to develop any machinery unless it has an immediate payoff. However, I am not at all adverse to developing some tools from scratch as long as they lead to something cool.

It sounds like your students are ideal for a baby differential geometry course. As a perk, you could develop from DG the spherical and hyperbolic geometry models. This can satisfy 1-6 handily.
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Ryan BudneyNov 1 '12 at 3:12

A polytopes course could also satisfy 1-6 handily, but there are fewer books for this than for differential geometry.
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Alexander WooNov 1 '12 at 3:30

I had a course much like what you describe, Andy, but at U.Vic I had a more diverse array of backgrounds than you'd (likely) encounter at Rice. Baby DG went over fine. I used Millman and Parker as my text.
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Ryan BudneyNov 1 '12 at 3:41

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Differential Geometry is a good idea, but we already have an undergraduate course in differential geometry, so I probably should do something else. Is there a good undergraduate level book on polytopes? That could be a lot of fun (and I might learn something too).
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Andy PutmanNov 1 '12 at 3:42

I graded Peter Shalen's such course at one time. I think they used Artin's Geometric Algebra. You might find an axiomatic treatment boring, but Shalen had a number of applications, one being digging a railroad tunnel, that fostered part of my desire to attend the course rather than grade it. If you can, you might email him for suggestions. Gerhard "Still Went The Algebraic Route" Paseman, 2012.10.31
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Gerhard PasemanNov 1 '12 at 3:54

11 Answers
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I wonder whether Igor Pak's "Lectures on Discrete and Polyhedral Geometry" might be appropriate as a textbook for an undergraduate geometry course. This is still in preliminary form, available on his website. In the introduction he describes a selection of topics from the book that could be used for a basic undergraduate course. There seem to be lots of exercises, and at a quick glance a lot of the topics look quite interesting. This whole subject is way outside my expertise, however, so I have no idea if the book would make a good basis for a course like the one you'll be teaching.

Concerning the fourth item in your list, I believe that axiomatic approach is not only boring, but (and it is more important) it is almost useless for further mathematical courses.

In my view the best geometry you can teach your first year undergraduates is the one based on modern treatment of linear algebra. The syllabus might look like this (it is based on the course I've taken in the recent years):

Such a course would give your students better understanding of the geometric nature of linear algebra (personally I think that the material one learns in a linear algebra course should be called "linear geometry"), it would show how modern mathematics simplifies classic material such as euclidean geometry and it would provide strong geometric basis for courses like algebraic geometry and topology (where familiarity with projective spaces helps a lot).

I suggest reading the preface to Dieudonne's book where he elaborates on these issues.

I would recommend Continuous Symmetry : From Euclid to Klein by Barker and Howe. I took the course as an undergraduate and enjoyed it very much. The first chapter gives an axiomatic treatment of geometry, and is meant to be a short part of the course. The rest of the book is a transformational approach to geometry, introducing isometries and similarities. Felix Klein's Erlanger Programm is the guiding principle for the course.

1) I like the idea of a course about polytopes. Few books but some are excellent: "Lecture on polytopes" by Ziegler or "Convex polytopes" by Grunbaum are the obvious choices.

2) A course about curves and surfaces + an introduction to manifolds should satisfy 1-6 without troubles. "Differential geometry of curves and surfaces" by Do Carmo is a very good book; there are plenty of excellent books about manifolds.

3) A basic course on algebraic varieties require the use of algebra and differential calculus and gives example of spaces with pathological spaces (i.e. non Hausdorff and/or with singularities)

4) I guessed you want a more modern geometry course but without leaving the view of the formation of high school teachers. Michèle Audin wrote a very good book about affine, projective, curves and surfaces. It is aimed to future (French) high school teachers. I guess the title is "Geometry" (it is "Géométrie" in the French version).

I don't know the curriculum of a typical American student so I hope my suggestions are still pertinent (especially the point 3).

Even for the Rice undergraduate, I think Ziegler or Grunbaum would be too much as a textbook. I do not mean to say their content could not be taught to undergraduates, only that the presentation would be too dense for them to follow.
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Alexander WooNov 1 '12 at 7:18

Have you seen the book "Geometry by Its History" by Ostermann/Wanner?It looks anything but boring.I liked it very much!It has plenty of interestig mathematics.Maybe you record the lectures and put them on net...

Last semester I taught (at Colby College) a geometry course based on two books: Bonahon's "Low dimensional geometry" and Schwartz's "Mostly Surfaces". Both are relatively inexpensive as far as textbooks go, so I could require both from the students. The students really enjoyed reading both books simultaneously as the authors have very different styles but some overlap of content. The students would certainly need to know some linear algebra in addition to multivariable calculus. The course was challenging, but reasonably successful at helping students develop some "geometric imagination" and proof-writing skills.

For a more applied course: What about Jean Gallier: Geometric Methods andApplications.

From the preface:

"Novelties: As far as we know, there is no fully developed modern exposition integrating the basic concepts of affine geometry, projective geometry, Euclidean geometry, Hermitian geometry, basics of Hilbert spaces with a touch of Fourier series, basics of Lie groups and Lie algebras, as well as a presentation of curves and surfaces both from the standard differential point of view and from the algorithmic point of view in terms of control points (in the polynomial and rational case).

I like Euclidean and Non-Euclidean Geometries: Development and History by Marvin J. Greenberg. I will warn you: it is certainly an axiomatic treatment. However, I really enjoyed the way that the book develops it. For example, the distinction between the axioms of a geometry and theorems you can prove about them, versus the models of geometry and their various properties, is clearly drawn. I dare say that, despite how advanced your undergraduates feel, they will learn a lot about the axiomatic method from this book. I recommend that you give it a look; even if it is not the primary textbook for the course, you can use it as a convenient source of motivation, problems, examples, and history. (There is a lot of history in this book, and many exercises.)

I'm a a fan off
Episodes in 19th and 20th Century Geometry [Ross Honsberger]
It may have an old-fashioned outlook, but you would have to have simply amazing undergraduate students for none of them to find this material a challenge.
I would supplement this with perhaps some chapters from Miles Reid's Undergraduate Algebraic Geometry (especially his treatment of the 27 lines on a cubic surface)
and perhaps Alain Connes' little paper on Morley's Theorem.

I realize that most people prefer more foundation/methodological approaches (comparisons of various axiomatic systems). My bias: I want undergraduates to see more phenomena. But for that other sort of course, I like the
Strasbourg Master Class on Geometry.